Question: All of the 5th grade teachers and students from Covington went on a field trip to an archaeology museum. Tickets were $$8.50$ each for teachers and $$4.00$ each for students, and the group paid $$62.00$ in total. The next month, the same group visited a natural history museum where the tickets cost $$34.00$ each for teachers and $$9.50$ each for students, and the group paid $$202.50$ in total. Find the number of teachers and students on the field trips.
Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${8.5x+4y = 62}$ ${34x+9.5y = 202.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-4$ ${-34x-16y = -248}$ ${34x+9.5y = 202.5}$ Add the top and bottom equations together. $ -6.5y = -45.5 $ $ y = \dfrac{-45.5}{-6.5}$ ${y = 7}$ Now that you know ${y = 7}$ , plug it back into $ {8.5x+4y = 62}$ to find $x$ ${8.5x + 4}{(7)}{= 62}$ $8.5x+28 = 62$ $8.5x = 34$ $x = \dfrac{34}{8.5}$ ${x = 4}$ You can also plug ${y = 7}$ into $ {34x+9.5y = 202.5}$ and get the same answer for $x$ ${34x + 9.5}{(7)}{= 202.5}$ ${x = 4}$ There were $4$ teachers and $7$ students on the field trips.